There are two regimes of interest for the study of treewidth in sparse random graphs. For the subcritical regime \(p \le d/n\)  with \(d<1\) , \(G(n, p)\)  has w.h.p. unicyclic connected components of size \(O(\log n)\)  [1]} and hence has treewidth at most 2. For the supercritical regime with \(p \ge d/n\)  and \(d>1\) , \(G(n,p)\)  has w.h.p. a giant component of size \(\Omega (n)\)  [1]} and determining the treewidth is more complicated. Kloks [3]} proved that the treewidth of \(G(n, d/n)\)  is \(\Omega (n)\)  w.h.p. for \(d \ge 2.36\) . His result was improved by Gao [4]} who showed that for \(d \ge 2.16\) , the treewidth of \(G(n,d/n)\)  is \(\Omega (n)\)  with high probability. Gao asked if his result can be strengthened to prove that \(G(n, d/n)\)  has treewidth linear in \(n\)  w.h.p. for any \(d>1\) ; this was later shown in in [5]}. A different and somewhat simplified proof establishing that the treewidth of \(G(n,d/n)\)  is \(\Omega (n)\)  w.h.p. was given in [6]}. Finally, the fine-grained behavior of treewidth of \(G(n, (1+\epsilon )/n)\)  was studied in [7]} where it was shown that for sufficiently small \(\epsilon \) , the treewidth of \(G(n, (1+\epsilon )/n)\)  is w.h.p.
\(\Omega \left(\frac{\epsilon ^3}{\log {1/\epsilon }}\right)n.\) 
